Invariants of plane curve singularities and Pl\"ucker formulas in positive characteristic

We study classical invariants for plane curve singularities $f\in K[[x,y]]$, $K$ an algebraically closed field of characteristic $p\geq 0$: Milnor number, delta invariant, kappa invariant and multiplicity. It is known, in characteristic zero, that $\mu(f)=2\delta(f)-r(f)+1$ and that $\kappa(f)=2\delta(f)-r(f)+\mathrm{mt}(f)$. For arbitrary characteristic, Deligne prove that there is always the inequality $\mu(f)\geq 2\delta(f)-r(f)+1$ by showing that $\mu(f)-\left( 2\delta(f)-r(f)+1\right)$ measures the wild vanishing cycles. By introducing new invariants $\gamma,\tilde{\gamma}$, we prove in this note that $\kappa(f)\geq \gamma(f)+\mathrm{mt}(f)-1\geq 2\delta(f)-r(f)+\mathrm{mt}(f)$ with equalities if and only if the characteristic $p$ does not divide the multiplicity of any branch of $f$. As an application we show that if $p$ is "big" for $f$ (in fact $p > \kappa(f)$), then $f$ has no wild vanishing cycle. Moreover we obtain some Pl\"ucker formulas for projective plane curves in positive characteristic.Comment: 15 pages; final version; to appear in the Annales de l'Institut Fourie

Volume form on moduli spaces of d-differentials

Given $d\in \mathbb{N}$, $g\in \mathbb{N} \cup\{0\}$, and an integral vector $\kappa=(k_1,\dots,k_n)$ such that $k_i>-d$ and $k_1+\dots+k_n=d(2g-2)$, let $\Omega^d\mathcal{M}_{g,n}(\kappa)$ denote the moduli space of meromorphic $d$-differentials on Riemann surfaces of genus $g$ whose zeros and poles have orders prescribed by $\kappa$. We show that $\Omega^d\mathcal{M}_{g,n}(\kappa)$ carries a canonical volume form that is parallel with respect to its affine complex manifold structure, and that the total volume of $\mathbb{P}\Omega^d\mathcal{M}_{g,n}(\kappa)=\Omega^d\mathcal{M}_{g,n}/\mathbb{C}^*$ with respect to the measure induced by this volume form is finite.Comment: Streamlined, minor corrections added, definition of the volume form independent of the choice of a d-th root of unit

Right unimodal and bimodal singularities in positive characteristic

The problem of classification of real and complex singularities was initiated by Arnol'd in the sixties who classified simple, unimodal and bimodal w.r.t. right equivalence. The classification of right simple singularities in positive characteristic was achieved by Greuel and the author in 2014. In the present paper we classify right unimodal and bimodal singularities in positive characteristic by giving explicit normal forms. Moreover we completely determine all possible adjacencies of simple, unimodal and bimodal singularities. As an application we prove that, for singularities of right modality at most 2, the $\mu$-constant stratum is smooth and its dimension is equal to the right modality. In contrast to the complex analytic case, there are, for any positive characteristic, only finitely many 1-dimensional (resp. 2-dimensional) families of right class of unimodal (resp. bimodal) singularities. We show that for fixed characteristic $p>0$ of the ground field, the Milnor number of $f$ satisfies $\mu(f)\leq 4p$, if the right modality of $f$ is at most 2.Comment: 19 page

The right classification of univariate power series in positive characteristic

While the classification of univariate power series up to coordinate change is trivial in characteristic 0, this classification is very different in positive characteristic. In this note we give a complete classification of univariate power series $f\in K[[x]]$, where $K$ is an algebraically closed field of characteristic $p>0$ by explicit normal forms. We show that the right determinacy of $f$ is completely determined by its support. Moreover we prove that the right modality of $f$ is equal to the integer part of $\mu/p$, where $\mu$ is the Milnor number of $f$. As a consequence we prove in this case that the modality is equal to the proper modality, which is the dimension of the $\mu$-constant stratum in an algebraic representative of the semiuniversal deformation with trivial section.Comment: 17 pages, final versio

Translation surfaces and the curve graph in genus two

Let $S$ be a (topological) compact closed surface of genus two. We associate to each translation surface $(X,\omega) \in \mathcal{H}(2)\sqcup\mathcal{H}(1,1)$ a subgraph $\hat{\mathcal{C}}_{\rm cyl}$ of the curve graph of $S$. The vertices of this subgraph are free homotopy classes of curves which can be represented either by a simple closed geodesic, or by a concatenation of two parallel saddle connections (satisfying some additional properties) on $X$. The subgraph $\hat{\mathcal{C}}_{\rm cyl}$ is by definition $\mathrm{GL}^+(2,\mathbb{R})$-invariant. Hence, it may be seen as the image of the corresponding Teichm\"uller disk in the curve graph. We will show that $\hat{\mathcal{C}}_{\rm cyl}$ is always connected and has infinite diameter. The group ${\rm Aff}^+(X,\omega)$ of affine automorphisms of $(X,\omega)$ preserves naturally $\hat{\mathcal{C}}_{\rm cyl}$, we show that ${\rm Aff}^+(X,\omega)$ is precisely the stabilizer of $\hat{\mathcal{C}}_{\rm cyl}$ in ${\rm Mod}(S)$. We also prove that $\hat{\mathcal{C}}_{\rm cyl}$ is Gromov-hyperbolic if $(X,\omega)$ is completely periodic in the sense of Calta. It turns out that the quotient of $\hat{\mathcal{C}}_{\rm cyl}$ by ${\rm Aff}^+(X,\omega)$ is closely related to McMullen's prototypes in the case $(X,\omega)$ is a Veech surface in $\mathcal{H}(2)$. We finally show that this quotient graph has finitely many vertices if and only if $(X,\omega)$ is a Veech surface for $(X,\omega)$ in both strata $\mathcal{H}(2)$ and $\mathcal{H}(1,1)$.Comment: 47 pages, 17 figures. Minor changes, some proofs improved. Comments welcome